Probabilistic graphical models in modern social network analysis

Abstract

The advent and availability of technology has brought us closer than ever through social networks. Consequently, there is a growing emphasis on mining social networks to extract information for knowledge and discovery. However, methods for social network analysis (SNA) have not kept pace with the data explosion. In this review, we describe directed and undirected probabilistic graphical models (PGMs), and highlight recent applications to social networks. PGMs represent a flexible class of models that can be adapted to address many of the current challenges in SNA. In this work, we motivate their use with simple and accessible examples to demonstrate the modeling and connect to theory. In addition, recent applications in modern SNA are highlighted, including the estimation and quantification of importance, propagation of influence, trust (and distrust), link and profile prediction, privacy protection, and news spread through microblogging. Applications are selected to demonstrate the flexibility and predictive capabilities of PGMs in SNA. Finally, we conclude with a discussion of challenges and opportunities for PGMs in social networks.

Appendix

1.1 Similarity between MNs and ERGMs

While MNs and ERGMs have been developed in different scientific domains, they both specify exponential family distributions. MN models treat social network nodes as random variables, and hence, their utility is most obvious in modeling processes on networks; ERGMs, on the other hand, have been conceptualized to model network formation, where it is the edge presence indicators that are treated as random variables (these random variables are dependent if their corresponding edges share a node). But in fact, this application-related difference in what to treat as random is not fundamental. This Appendix works to more rigorously disclose the similarity between MNs and ERGMs by re-defining an ERGM as a PGM. We begin, however, by reviewing the branch of literature devoted exclusively to ERGMs.

Similar to MNs, a well-discussed problem of ERGMs for analyzing social networks is related to the challenge of parameters estimation (Robins et al. 2007b) due to the lack of enough observed data. Robins et al. (2007b) outline this and some other problems associated with ERGMs, e.g., degeneracy in model selection and bimodal distribution shapes (see also Handcock et al. 2003; Rinaldo et al. 2009; Snijders et al. 2006; Handcock et al. 2006).

The roots of ERGMs in the Principle of Maximum Entropy (Park and Newman 2004) and the Hammersley–Clifford theorem have been previously pointed out (Robins et al. 2001; Goldenberg et al. 2010). Here, we illustrate how MNs and ERGMs are similar in terms of the form and structure using most popular significant statistics in ERGMs; under the assumption of Markov dependence, for a given social network, one can build a corresponding Markov network via the following conversion: (1) each node in the Markov network will correspond to an edge in the social network [Fienberg called this construct a “usual graphical model” for ERGMs (Fienberg 2012)], (2) when two edges share a node in the social network, a link will be built between two corresponding nodes in the Markov network.

Corresponding to each possible edge in a social network, a node in an MN network is introduced; note the difference between the original social network and the MN network—they are not the same! Consider an ERGM with the significant statistics including the number of edges, $f_{1}(y)$, the number of k-stars, $f_{i}(y),i =2,\ldots ,N-1$ and the number of triangles, $f_{N}(y)$. In an MN, a maximum Entropy (maxent) model proposes the following form for the internal energy of the system, $E_{c}(x)= -\sum _{i}{\alpha _{ci}g_{ci}}$. Define, $g_{ci}$ as $i^{th}$ feature of clique $c \in \varOmega$ and $\alpha _{ci}$ is its corresponding weight in G. Thus, $\psi _{c}(x)=\exp \{\beta _c\sum _{i=1}^N{\alpha _{ci}g_{ci}}\}$. Since there are too many parameters in the MN, they can be deducted by imposing homogeneity constraints similar to that of ERGMs (Robins et al. 2007a). Before imposing such constraints, these following facts are required.

It is straightforward to demonstrate that G encompasses cliques of size $\{3, \ldots ,N-1\}$. In addition, all substructure in $G_s$ can be redefined by features in G. Considering these points, we can rewrite the joint probability of all variables represented by the MN, P(X), as follows:

$$\begin{aligned} P(X)&=\frac{1}{Z(\alpha )}\prod _{c=1}^C{\exp \left( \beta _c\sum _{i=1}^N{\alpha _{ci}g_{ci}}\right) } \\ &= \frac{1}{Z(\alpha )}\exp \left( \sum _{c=1}^C{\beta _c\sum _{i=1}^N{\alpha _{ci}g_{ci}}}\right) . \end{aligned}$$

(6)

In (4), $Z(\alpha )$ is the partition function which is a function of parameters. The homogeneity assumption, here, means $\alpha _{ci}=\theta _i'\; \forall \; c=1,\ldots , C$; then P(X) is:

$$\begin{aligned} P(X)=\frac{1}{Z(\theta ')}\exp \left( \sum _{i=1}^N{\theta _i'\sum _{c=1}^C{\beta _cg_{ci}}}\right) . \end{aligned}$$

(7)

In (5), let’s $Z'=Z(\theta ')$. In addition, we assume that $\sum _{c=1}^C{\beta _cg_{ci}}$ represented by $f_i'$, means that substructures i in all cliques c are added up by weight $\beta _c$. Finally, if we replace $f_i'$ in (5):

$$\begin{aligned} P(X)=\frac{1}{Z'}\exp \left( \sum _{i=1}^N{\theta _i'f_i'}\right) . \end{aligned}$$

(8)

Comparing $P(Y=y)$ and (4) confirms that ERGMs and MNs are similar and under the following conditions they are identical:

• 1. $\theta _i=\theta _i'$,

• 2. $f_i=f_i'=\sum _{c=1}^C{\beta _cg_{ci}}$.

The following Numerical Example (the same example in the ERGM section) depicts similarities between ERGMs and MNs. The social network has five actors, $N=5$ (Fig. 8). Considering Markov dependency assumption, there exists an unique corresponding Markov network shown in Fig. 9 with 10 nodes. There are 15 cliques (so-called factors) of size three or four,

$$\begin{aligned} {\varvec{\Phi }}=\{\phi _1(y_{12},y_{13},y_{14},y_{15}), \ldots ,\phi _{15}(y_{24},y_{45},y_{25})\}. \end{aligned}$$

Fig. 9

A social network with five actors (left) and its corresponding Markov network (right)

As already mentioned, the joint probability function of all variables in each clique is proportional to the internal energy. For instance:

$$\begin{aligned} \phi _{1}(x)=\frac{1}{\lambda }\exp \{-\beta _1 E_c(y_{12},y_{13},y_{14},y_{15})\}, \end{aligned}$$

where $E_{1}(x)=-\sum _{i}{\alpha _{ci}g_{ci}}$ and $\lambda$ is the distribution parameter. This simple example shows that how ERGMs and MNs are the same in terms of the underlying concept and the expressed probability distribution.